Do we need non-linear activation function in neural networks whose task isn't classification?
Yes, definitely.
In the simplest example, predicting an output value for a time series is classification. You take in the previous time steps and classify what is the most likely next value. You could do this with a RNN (Recurrent Neural Network) for example.
If the activation functions are all linear, the nerual network is just a glorified linear regression. Think of it like this: a neural network is trying to approximate a complicated function in $n$ dimensional space. It does this by combining operations on a series of known functions, to get a resultant function that hopefully mimics the desired function. The issue with combining linear functions is the only thing you'll ever get at the end is a linear function.
As a concrete example, try and approximate the function $y = x^3 + x^2 -x -1$ by adding a series of linear functions together. You'll find pretty quickly this is useless. However, if you use a non-linear function, such as a ReLU (Rectified Linear Unit) you can quite easily approximate this function. See this implementation on desmos.
If a problem has any sort of complexity to it, the function it follow is likely incredibly complicated, and futile to approximate using linear equations.